Extremum Monte Carlo Filters: Signal Extraction via Simulation and Regression

We introduce a novel simulation-based method for signal extraction in a general class of state space models. It can be used to estimate time-varying conditional means, modes, and quantiles, and to predict latent variables or forecast observations. The method consists of generating artificial datasets from the model and estimating the quantities of interest via extremum estimation. The approach is broadly applicable and its implementation is straightforward. The method is suited for signal extraction in cases of long time series, missing data, or high-dimensionality. Furthermore, we demonstrate its use in real-time filtering, where most of the computations can be performed in advance, and in fixed-interval smoothing. Conditions for the stability and convergence of the filtering method are discussed, and its key properties are illustrated by various applications, including nonlinear and high-dimensional models.